This is a question from an old qualifying exam.

Let $f=u+iv$ be a holomorphic function on the open unit disc into the complex plane. Among all holomorphic functions $f$ with $v(0) = 0$ and $|u|<1$ what is the largest possible value of $v(1/2)$.


My initial idea was to let $g = e^f$ so that $|g(z)| = e^{u(z)}$ and $$r = \frac{z-e^{u(0)}}{e-\frac{1}{e}e^{u(0)}z}.$$ Then letting $h = r\circ g$, which is a holomorphic map from the unit disc into the unit disc with $h(0) = 0$, we can apply Schwarz's Lemma to conclude that $|h(1/2)| \leq 1/2$.

This seems to be a standard way to approach these types of problems, however I wasn't able to find an upper bound on $v(1/2)$ using this method.

  • 1
    $\begingroup$ Here is a way to bound $|g(1/2)|$, this implies the bound for $|v(1/2)|$ which I think will be sharp. $\endgroup$ – user100000 Oct 17 '13 at 1:52

Schwarz lemma states that the equality of $|f(z)|\le |z|$ holds iff $f$ is a rotation. Use this.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.